We describe the space-time finite element techniques we developed for computation of fluid-structure interaction (FSI) problems. Among these techniques are the Deforming-Spatial-Domain/Stabilized Space-Time (DSD/SST) formulation and its special version, and the mesh update methods, including the Solid-Extension Mesh Moving Technique (SEMMT). Also among these techniques are the block-iterative, quasidirect and direct coupling methods for the solution of the fully-discretized, coupled fluid and structural mechanics equations. We present some test computations for the mesh moving techniques described. We also present numerical examples where the fluid is governed by the Navier-Stokes equations of incompressible flows and the structure is governed by the membrane and cable equations. Overall, we demonstrate that the techniques we have developed have increased the scope and accuracy of the methods used in computation of FSI problems.